Low sidelobe waveform design method and system based on Riemannian manifold

Abstract

The invention discloses a low sidelobe waveform design method and system based on a Riemannian manifold, and the method comprises the steps: introducing a peak-to-average power ratio as a constraint condition based on an MIMO radar system, and constructing an unconstrained optimization problem of the manifold; introducing a tangent plane, and performing iterative search on the unconstrained optimization problem of the flow shape to obtain new data points on the tangent plane; and constructing a remapping function to map the new data points on the tangent plane into the back flow form, thereby realizing the design of the minimum low-sidelobe waveform. According to the embodiment of the invention, a manifold structure in Riemannian geometry can be introduced into a non-convex waveform design optimization problem, the calculation complexity is reduced, and the algorithm convergence speed and the solving precision are effectively improved. The method can be widely applied to the technical field of radar waveform design.

Description

Technical Field

[0001] The present application relates to the technical field of radar waveform design, and in particular to a low-sidelobe waveform design method and system based on Riemannian manifold. Background Art

[0002] Modern radar systems face urgent demands for multi-target resolution, spectrum sharing, and the development of MIMO radars in complex electromagnetic environments, placing higher demands on the design of low-sidelobe waveform sets. Traditional waveform design methods are primarily based on Euclidean space optimization theory, including convex optimization methods (such as semidefinite programming), loop algorithms (CAN), and stochastic optimization algorithms. While these methods can achieve sidelobe suppression of the order of -40 dB in specific scenarios, they generally suffer from high computational complexity, prone to local optimality, and difficulty maintaining orthogonality in the waveform set. Although low-sidelobe waveform design using neural networks can effectively accelerate convex optimization algorithms, their optimization results are highly dependent on the setting of the regularization term in the loss function. This makes it difficult to ensure that the output waveforms strictly meet the constraints in complex MIMO waveform design scenarios, such as those with constant modulus and orthogonality constraints. Both traditional waveform optimization methods and those based on neural networks rely on specific signal models, which may deviate from the optimal waveform space.

[0003] In summary, the technical problems existing in the relevant technologies need to be improved. Summary of the Invention

[0004] The main purpose of the embodiments of this application is to propose a low-sidelobe waveform design method and system based on Riemannian manifolds, which can introduce the manifold structure in Riemannian geometry into the non-convex waveform design optimization problem, reduce the computational complexity, and thus effectively improve the algorithm convergence speed and solution accuracy.

[0005] To achieve the above objectives, an embodiment of the present application provides a low sidelobe waveform design method based on Riemannian manifolds, the method comprising:

[0006] Based on a MIMO radar system, the peak-to-average power ratio is introduced as a constraint to construct an unconstrained optimization problem on a manifold, which is a high-dimensional hypersphere in complex space with a preset radius.

[0007] A tangent plane is introduced, and an unconstrained optimization problem of the manifold is iteratively searched to obtain new data points on the tangent plane;

[0008] A remapping function is constructed to map the new data points on the tangent plane back to the manifold to achieve a minimized low sidelobe waveform design.

[0009] In some embodiments, the MIMO radar system introduces a peak-to-average power ratio as a constraint to construct an unconstrained optimization problem on a manifold, including:

[0010] Based on a MIMO radar system, a transmission data matrix of the MIMO radar system is obtained and constant modulus constraints are performed to obtain an ideal transmission data matrix;

[0011] Defining a minimization criterion of a correlation function of the ideal transmission data matrix as an optimization criterion for waveform design, defining a norm of the ideal transmission data matrix as a measure for minimizing a peak sidelobe level, and constructing a peak sidelobe level function for minimizing a transmission waveform;

[0012] The peak sidelobe level function of the minimized transmission waveform is transformed by using the peak-to-average power ratio as a constraint condition to obtain a constrained nonlinear optimization problem;

[0013] The feasible solution of the peak sidelobe level function of the minimized transmission waveform is defined in a manifold obtained by multiplying a plurality of smooth Riemannian manifolds, and the constrained nonlinear optimization problem is converted into an unconstrained optimization problem of the manifold.

[0014] In some embodiments, the expression of the peak sidelobe level function of the minimized transmission waveform is specifically as follows:

[0015]

[0016] In the above formula, f(X) represents the norm of the ideal transmission data matrix, X represents the ideal transmission data matrix, and w a 、w c represents the weight coefficient, represents the autocorrelation function level of all antennas, represents the cross-correlation function level between all antennas, N k represents a matrix that changes with the change of the delay variable k, k represents the delay variable, x i 、x j Represents the transmitted waveform sequence, M represents the number of transmitting antennas, and N represents the number of waveform samples collected by each antenna. Represents x i The conjugate transpose of , p represents the power of the norm.

[0017] In some embodiments, the expression of the unconstrained optimization problem of the manifold is specifically as follows:

[0018]

[0019] In the above formula, represents the manifold, X represents the ideal emission data matrix, f(X) represents the norm of the ideal emission data matrix, represents an element in the manifold, x m represents the sequence transmitted by the mth antenna, represents the complex space, N represents the number of waveform samples collected by each antenna, ε represents the threshold corresponding to different levels of peak-to-average power ratio, m represents the mth antenna, and M represents the number of transmitting antennas.

[0020] In some embodiments, the introducing of the tangent plane and performing an iterative search on the unconstrained optimization problem of the manifold to obtain new data points on the tangent plane includes:

[0021] A tangent plane is defined, and the unconstrained optimization problem of the manifold is linearized through the tangent plane to obtain data points of the tangent plane, where the tangent plane is a vector space composed of tangent vectors of all manifolds passing through the points;

[0022] On the tangent plane, calculating the current Riemann gradient for the data point to determine the fastest descent direction;

[0023] Determine a direction search function based on the current Riemann gradient and determine a step size using an Armijo line search algorithm;

[0024] The direction search function is used to convert the current Riemann gradient into a specific search direction to search the unconstrained optimization problem of the manifold, and iterative update is performed in combination with the step size to obtain new data points on the tangent plane.

[0025] In some embodiments, the expression for determining the step size by the Armijo line search algorithm is specifically as follows:

[0026]

[0027] In the above formula, X k represents the new data point on the tangent plane, α k Indicates the current step size, D k Indicates the current search direction. represents the directional derivative, G k Conjugate transpose, G k represents the Riemann gradient of the kth iteration, σ represents the update coefficient, X k represents the waveform matrix of the kth iteration, f(X k ) represents the expression of the peak sidelobe level function, f(X k +α k D k ) represents the updated peak sidelobe level.

[0028] In some embodiments, the expression of the direction search function is specifically as follows:

[0029] D k =H(X k )=-G k +β k ·D k-1

[0030] In the above formula, D k Indicates the current search direction, H(X k ) represents the search function, G k represents the Riemann gradient, β k Denotes the coefficient, D k-1 Indicates the last search direction.

[0031] In some embodiments, constructing a remapping function to map the new data points on the tangent plane back to the manifold to achieve a minimized low sidelobe waveform design includes:

[0032] defining a remapping function to obtain the norm of the new data point on the tangent plane;

[0033] Normalizing the new data points on the tangent plane using the norm to obtain normalized new data points;

[0034] The normalized new data points are input into the remapping function, multiplied by the radius of the high-dimensional hypersphere, and the new data points on the tangent plane are mapped back to the manifold to achieve minimized low sidelobe waveform design.

[0035] In some embodiments, the expression of the remapping function is specifically as follows:

[0036]

[0037] In the above formula, represents the remapping function, X k represents the new data point on the tangent plane, α k Indicates the current step size, D k Indicates the current search direction. represents the radius of the high-dimensional hypersphere, ||·|| F Represents the norm.

[0038] To achieve the above objectives, another aspect of the present application provides a low sidelobe waveform design system based on Riemannian manifolds, the system comprising:

[0039] The first module is used to construct an unconstrained optimization problem of a manifold based on a MIMO radar system by introducing the peak-to-average power ratio as a constraint condition. The manifold is a high-dimensional hypersphere in a complex space with a preset radius length.

[0040] The second module is used to introduce a tangent plane, perform an iterative search on the unconstrained optimization problem of the manifold, and obtain new data points on the tangent plane;

[0041] The third module is used to construct a remapping function to map the new data points on the tangent plane back to the manifold to achieve minimized low sidelobe waveform design.

[0042] The embodiments of the present application include at least the following beneficial effects: The present application provides a low sidelobe waveform design method and system based on Riemann manifolds. The scheme introduces the peak-to-average power ratio as a constraint condition based on the MIMO radar system, constructs an unconstrained optimization problem of the manifold, and uses the Riemann manifold optimization method to transform the original NP-hard constrained problem into an unconstrained problem in the manifold space, effectively avoiding the computational complexity brought by the constraint processing. The tangent plane is further introduced, and the unconstrained optimization problem of the manifold is iteratively searched to obtain new data points on the tangent plane. The Riemann manifold is used to search for waveforms that meet the requirements from a specific structure, thereby improving the convergence speed and solution accuracy of the algorithm. Finally, a remapping function is constructed to map the new data points on the tangent plane back to the manifold to achieve minimized low sidelobe waveform design. It is possible to design a low-correlation sidelobe waveform matrix with good orthogonal characteristics under the specified peak-to-average power ratio constraint, reduce computational complexity, and effectively improve the convergence speed and solution accuracy of the algorithm. BRIEF DESCRIPTION OF THE DRAWINGS

[0043] Figure 1 This is a flow chart of a low sidelobe waveform design method based on Riemannian manifolds provided in an embodiment of the present application;

[0044] Figure 2 Schematic diagram of a low sidelobe waveform design system based on Riemannian manifolds provided in an embodiment of the present application;

[0045] Figure 3 1 is a schematic diagram of the structure of the MIMO radar system provided in an embodiment of the present application;

[0046] Figure 4 It is a schematic diagram of a cutting plane provided in an embodiment of the present application;

[0047] Figure 5 Schematic diagram of the iteration of the optimization algorithm based on Riemannian manifold provided in the embodiment of the present application;

[0048] Figure 6 This is a schematic diagram of the process of low sidelobe waveform design provided by an embodiment of the present application;

[0049] Figure 7 Schematic diagram of the change of the first-order approximation error with the step size provided in an embodiment of the present application;

[0050] Figure 8This is a diagram showing a curve of the loss function value changing with the number of iterations provided in an embodiment of the present application;

[0051] Figure 9 This is a schematic diagram of the effect of the number of iterations on ACF provided in an embodiment of the present application;

[0052] Figure 10 Schematic diagram of the effect of the convergence tolerance of the gradient norm on PSL provided in an embodiment of the present application;

[0053] Figure 11 This is a schematic diagram of the effect of the minimum constraint step size on PSL provided in an embodiment of the present application;

[0054] Figure 12 This is a schematic diagram of correlation function results under different PAPRs when N=50 provided in an embodiment of the present application;

[0055] Figure 13 1 is a schematic diagram of correlation function results under different PAPRs when N=100 provided in an embodiment of the present application;

[0056] Figure 14 This is a schematic diagram of correlation function results under different PAPRs when N=200 provided in an embodiment of the present application;

[0057] Figure 15 Schematic diagram of output waveform modulus values ​​under different PAPRs provided in an embodiment of the present application. DETAILED DESCRIPTION

[0058] In order to make the purpose, technical solutions and advantages of the present application clearer, the present application is further described in detail below with reference to the accompanying drawings and examples. It should be understood that the specific embodiments described herein are only used to explain the present application and are not intended to limit the present application. When the following description refers to the drawings, unless otherwise indicated, the same numbers in different drawings represent the same or similar elements. The embodiments described in the following exemplary embodiments do not represent all embodiments consistent with the embodiments of the present application. They are merely examples of systems and methods consistent with some aspects of the embodiments of the present application as detailed in the appended claims.

[0059] It will be understood that the terms "first", "second", etc. used in this application may be used herein to describe various concepts, but unless otherwise specified, these concepts are not limited by these terms. These terms are only used to distinguish one concept from another. For example, without departing from the scope of the embodiments of the present application, the first information may also be referred to as the second information, and similarly, the second information may also be referred to as the first information. Depending on the context, the words "if" and "if" as used herein may be interpreted as "at the time of" or "when" or "in response to determining".

[0060] The terms "at least one", "plurality", "each", "any", etc. used in this application include "at least one", "two" or more, "plurality" or "each", "any" or "any one", "each" or "any one" in the context of the present invention, and "at least one" or "at least one" includes one, two or more, "plurality" or "any one" includes two or more, "each" or "each one" in the context of the present invention, and "any" or "any one

[0061] Unless otherwise defined, all technical and scientific terms used herein have the same meaning as commonly understood by those skilled in the art to which this application pertains. The terms used herein are for the purpose of describing the embodiments of this application only and are not intended to limit this application.

[0062] First of all, it should be noted that there are some deficiencies in the relevant technology, such as:

[0063] 1) The Circular Optimization Algorithm (CAN), a classic phase optimization method, approaches ideal autocorrelation characteristics through iterative fast Fourier transforms. This technique first randomly initializes the phase sequence and then iteratively executes three core steps: calculating the autocorrelation function of the current waveform, forcing the sidelobes outside a specified distance window to zero, and then updating the phase value through an inverse transform. After 50-100 iterations, the algorithm typically achieves an optimized result with sidelobe levels below -40dB. However, this frequency-domain alternating projection mechanism has two inherent drawbacks: First, its iteration efficiency scales nonlinearly with waveform length, significantly increasing computational latency when processing waveforms with more than 128 points. Second, the algorithm framework is difficult to scale to multiple waveforms for joint optimization. When processing more than four waveforms, the nested iterations required to maintain orthogonality lead to an explosive increase in computational complexity. Furthermore, the algorithm is extremely sensitive to quantization errors in the hardware implementation. In practical deployments, 8-bit quantization can degrade theoretical performance by over 30%.

[0064] 2) Peak Sidelobe Optimization Iteration (POCA) algorithm. The POCA method is an optimization algorithm specifically designed for periodic radar waveforms. Its core concept is to iteratively adjust the phase coding sequence so that the waveform maintains extremely low autocorrelation sidelobes during periodic repetition. The method first constructs a random phase sequence as an initial solution. It then iteratively transforms between the frequency and time domains. After each calculation of the waveform's periodic autocorrelation function, the sidelobes outside the mainlobe region are forced to zero, and the waveform parameters are then updated through phase reconstruction. After dozens of iterations, the algorithm can generate a constant envelope signal with sidelobe levels below -45dB, making it particularly suitable for applications such as continuous wave radar that require periodic waveform transmission. However, the POCA method has two inherent limitations: First, its excellent sidelobe suppression performance strictly depends on periodic boundary conditions. When applied to non-periodic scenarios such as pulse radar, the actual sidelobe level deteriorates by 3-5dB. Second, the algorithm's high computational complexity makes it difficult to process long waveforms. When the code length exceeds 256 bits, the optimization time increases nonlinearly, severely restricting its application potential in real-time systems.

[0065] 3) Neural network methods: The waveform set design method based on neural networks constructs an end-to-end waveform optimization framework through deep generative models (such as generative adversarial networks and variational autoencoders). This method encodes the requirements of radar waveforms such as autocorrelation sidelobe suppression, cross-correlation orthogonality constraints, and constant mode characteristics into the loss function of the neural network, and uses the gradient backpropagation mechanism to automatically learn the optimal waveform parameter space. The typical implementation includes two stages: offline training and online generation. The offline stage uses a large amount of simulated data to train the network to learn the implicit expression of the waveform's low-sidelobe characteristics; the online stage uses a single forward calculation to generate a waveform set that meets multiple objective constraints in real time, reducing the computational time by 2-3 orders of magnitude compared to traditional iterative algorithms.

[0066] Therefore, traditional waveform design methods are primarily based on Euclidean space optimization theory, including convex optimization methods (such as semidefinite programming), loop algorithms (CAN), and stochastic optimization algorithms. While these methods can achieve sidelobe suppression of the order of -40 dB in specific scenarios, they generally suffer from high computational complexity, prone to falling into local optimality, and difficulty maintaining orthogonality in waveform sets. In particular, when the waveform dimension N > 100, the computational time of traditional methods increases exponentially, severely restricting real-time applications. In recent years, manifold optimization theory has begun to be introduced into the field of radar waveform design. Existing research has shown that Stiefel manifolds and Grassmann manifolds can partially improve optimization performance, but they still fail to fully utilize the Riemannian geometry properties of the parameter space and have significant deficiencies in convergence speed and multi-constrained joint optimization. Furthermore, existing technologies lack an effective mathematical representation of the coupling relationship between the autocorrelation and cross-correlation characteristics of waveform sets, resulting in performance limitations in scenarios such as MIMO radar that require the coordinated optimization of multiple waveforms.

[0067] With the rapid development of deep learning technology, neural network-based radar waveform design methods have gradually become a research hotspot. Traditional neural network methods demonstrate significant advantages in reducing computational complexity by building deep generative models (such as generative adversarial networks and variational autoencoders) to learn optimal waveform feature representations. Research has shown that an end-to-end neural network architecture can transform the traditional iterative optimization process into a single forward computation, reducing computational time by 2-3 orders of magnitude while maintaining -45dB sidelobe suppression. However, existing neural network methods still have several key limitations: First, the black-box nature of the model makes it difficult to strictly enforce physical constraints on waveform parameters (such as constant modulus constraints); second, network training requires massive amounts of labeled data, while obtaining samples of real radar environments is difficult; and finally, existing methods lack the ability to model the orthogonality of waveform sets, making it difficult to achieve coordinated optimization of autocorrelation and cross-correlation characteristics. Recent work has begun to explore hybrid neural network architectures that embed physical knowledge (such as coupled phase encoding modules), but these methods still face the challenge of balancing adaptability to dynamic environments with computational efficiency, particularly when deployed on embedded platforms, due to performance losses caused by model compression.

[0068] Manifold-based methods directly search for waveforms that meet requirements on a specific manifold structure, resulting in more accurate optimization results. In recent years, optimization methods based on Riemannian manifolds have garnered widespread attention in MIMO radar waveform design. The Riemannian manifold is a core concept in differential geometry. It defines a Riemannian metric on a smooth manifold, enabling the solution of radar waveforms on this manifold structure. The solution space for radar waveforms is no longer the traditional Euclidean space, but a nonlinear space composed of Riemannian manifolds with a specific geometric structure.

[0069] In view of this, an embodiment of the present application provides a low sidelobe waveform design method based on Riemann manifold. In the constant modulus constrained waveform design, the optimal waveform set can be regarded as an optimization problem on a complex sphere or a complex Stiffel manifold. Using the Riemann manifold optimization method, the original NP-hard constrained problem can be transformed into an unconstrained problem in the manifold space, thereby effectively avoiding the computational complexity brought by constraint processing, while effectively improving the convergence speed and solution accuracy of the algorithm.

[0070] Reference Figure 1 , Figure 1 A flowchart of a low sidelobe waveform design method based on Riemannian manifold is provided in an embodiment of the present invention, referring to Figure 1 , the method comprises the following steps:

[0071] S100, based on a MIMO radar system, introduces the peak-to-average power ratio as a constraint to construct an unconstrained optimization problem on a manifold, which is a high-dimensional hypersphere in complex space with a preset radius length;

[0072] It should be noted that, in some embodiments, step S100 may include steps S110 to S140:

[0073] S110, based on the MIMO radar system, obtaining a transmission data matrix of the MIMO radar system and performing constant modulus constraints to obtain an ideal transmission data matrix;

[0074] In this embodiment, if Figure 3 As shown, consider a MIMO radar system with M transmitting antennas and N waveform samples collected by each antenna. For the mth antenna in this MIMO radar system, it corresponds to a transmitting sequence transmitting waveform vector x m =[x m (0),x m (1),…,x m (N-1)] T (m=1,2,…,M). Then the transmission data matrix X of the entire MIMO radar system can be expressed as:

[0075] X N×M =[x1,x2,…,x M ]

[0076] Constant modulus constraints are extremely important for waveform design. When a waveform significantly deviates from these constraints, output signal distortion may occur due to the limited dynamic range of the power amplifier. Therefore, for an ideal transmit data matrix X, it should satisfy the following constraints:

[0077]

[0078] That is, the modulus value of each element of the transmitted data matrix X is 1.

[0079] S120, defining a minimization criterion of a correlation function of an ideal transmission data matrix as an optimization criterion for waveform design, defining a norm of the ideal transmission data matrix as a measure for minimizing a peak sidelobe level, and constructing a peak sidelobe level function for minimizing a transmission waveform;

[0080] In the embodiment of the present invention, the transmission waveform sequence x i and the emission waveform sequence x j The expression of the correlation function between is as follows:

[0081]

[0082] Due to the conjugate symmetry of the correlation function, only the delay variable k ranging from 0 to N-1 needs to be considered. Using the correlation function PSL minimization criterion as the optimization criterion for waveform design, and using the norm f(X) of the transmit data matrix X to represent PSL, the expression for minimizing the peak sidelobe level function of the transmit waveform is as follows:

[0083]

[0084] In the above formula, f(X) represents the norm of the ideal transmission data matrix, X represents the ideal transmission data matrix, and w a 、w c represents the weight coefficient, represents the autocorrelation function level of all antennas, represents the cross-correlation function level between all antennas, N k represents a matrix that changes with the change of the delay variable k, k represents the delay variable, x i 、x j Represents the transmitted waveform sequence, M represents the number of transmitting antennas, and N represents the number of waveform samples collected by each antenna. Represents x i The conjugate transpose of , p represents the power of the norm.

[0085] Among them, the first represents the autocorrelation function level of all antennas, and the second term Represents the level of cross-correlation function between all antennas, corresponding to the weight coefficient w a and w c . N k It is a matrix that changes with the change of the delay variable k. The mathematical expression is as follows:

[0086]

[0087] In the above formula, N k Represents a matrix that changes with the delay variable k.

[0088] S130, converting the peak sidelobe level function of the minimized transmission waveform using the peak-to-average power ratio as a constraint condition to obtain a constrained nonlinear optimization problem;

[0089] S140. Define a feasible solution for the function of minimizing the peak sidelobe level of the transmitted waveform in a manifold obtained by multiplying several smooth Riemannian manifolds, and convert the constrained nonlinear optimization problem into an unconstrained optimization problem of the manifold.

[0090] In some specific embodiments, when p approaches +∞, the norm f(X) approaches the PSL of the autocorrelation function of X. Since the constant modulus constraint is the most stringent PAPR constraint, in order to consider the impact of modulus constraints of different levels on the solution to the optimization problem, PAPR is used as a constraint to minimize the waveform's autocorrelation PSL. The mathematical expression of the optimization problem is as follows:

[0091]

[0092] Where ||·||2 is the Euclidean norm, ε corresponds to the threshold of different PAPR levels, and 1≤ε≤N. Assuming S is the solution space of the mathematical expression of the above optimization problem, the optimization problem is simplified to the following form, which is expressed as follows:

[0093] min X∈S f(X)

[0094] If the mathematical expression of the above optimization problem is an unconstrained optimization problem, since the emission data matrix X belongs to the N×M dimensional complex space and and real space are isomorphic, that is It can be seen that the solution to the unconstrained optimization problem belongs to a linear space. However, the series of PAPR constraints in the mathematical expression of the above optimization problem are non-convex. These constraints will cause the simplified solution space S to become a nonlinear space, making the entire optimization problem non-convex and NP-hard. In this case, traditional optimization algorithms based on Euclidean space (such as gradient descent and Newton's method) are no longer applicable, so new methods are needed. Due to the presence of PAPR constraints in the mathematical expression of the optimization problem, the optimization problem has a manifold structure. Therefore, to avoid using traditional optimization algorithms in Euclidean space, we consider introducing a manifold structure and thus transforming the problem into a manifold optimization problem.

[0095] For the sequence x transmitted by the mth antenna m , assuming that the maximum emission energy at each sampling moment is 1, that is Then in order to satisfy the PAPR constraint in the mathematical expression of the optimization problem, x m The feasible solution is limited to a complex space embedded in N dimensions. Smooth Riemannian manifold In the expression:

[0096]

[0097] From the expression of smooth Riemann manifold, we can see that the manifold The radius is Since the metric of this manifold can be defined by the inner product in complex space, this manifold also belongs to the Riemannian manifold.

[0098] Combined with the research content, it can be seen that by designing the parameters of the waveform, the transmission waveforms of different antennas in the MIMO radar system can have good orthogonality. Therefore, the feasible solution of X in the mathematical expression of the optimization problem is defined on the manifold obtained by multiplying M smooth Riemann manifolds. , the expression is as follows:

[0099]

[0100] The original nonlinear constrained optimization problem is then transformed into an unconstrained optimization problem in the manifold M:

[0101]

[0102] By manifold From the expression of There are M submanifolds The Cartesian product of Inherits the geometric structure of each submanifold: Since each submanifold is a sphere, and the sphere itself is a smooth manifold, It is also a smooth manifold, and similarly possesses a local Euclidean structure. This local Euclidean structure facilitates linearization of the manifold using tangent planes. Unlike optimization in traditional Euclidean space, this can be directly applied to smooth surfaces like Riemannian manifolds.

[0103] When optimizing with traditional optimization methods, the updated gradient often causes new data points to fall outside the manifold structure, leading to premature termination of the optimization iteration. Therefore, a tangent plane is introduced to the Riemannian manifold to minimize the deviation of new data points generated by the updated gradient from the manifold itself, thereby obtaining a new estimate and iterating.

[0104] S200, introducing a tangent plane, performing an iterative search on the unconstrained optimization problem of the manifold, and obtaining new data points on the tangent plane;

[0105] It should be noted that, in some embodiments, step S200 may include steps S210 to S240:

[0106] S210, defining a tangent plane, linearizing the unconstrained optimization problem of the manifold through the tangent plane, and obtaining data points of the tangent plane, where the tangent plane is a vector space composed of tangent vectors of all manifolds passing through the points;

[0107] In this embodiment, for the manifold For a data point x on Defined as all manifolds passing through point x The vector space composed of the tangent vectors of Figure 4 The tangent space provides a direction for data updates. When points on the manifold are updated along the direction of the tangent space, if the step size is small enough and the first-order Taylor expansion is satisfied, it can be assumed that the updated data points are still on the manifold.

[0108] S220, on the tangent plane, calculating the current Riemann gradient for the data point to determine the fastest descent direction;

[0109] In this embodiment, according to the theory of Riemannian geometry, the manifold is a set of points, and all points on the manifold have a Euclidean structure in the tangent space. Therefore, similar to using Euclidean gradients in linear space to update and optimize data, Riemannian gradients are used for optimization in Riemannian manifolds. Suppose the function to be optimized f(x) is at a point x on the manifold. k The value of f(x k ), then the Riemann gradient corresponding to this point can be expressed as This is a definition on the tangent plane. The Riemann gradient is the fastest descent direction of f(x) on the tangent plane, and is also the projection of the Euclidean gradient on the tangent plane.

[0110]

[0111] in is the orthogonal projection operator.

[0112] S230, determining a direction search function according to the current Riemann gradient and determining a step size by an Armijo line search algorithm;

[0113] S240. Use the direction search function to convert the current Riemann gradient into a specific search direction to search the unconstrained optimization problem of the manifold, and perform iterative updates in combination with the step size to obtain new data points on the tangent plane.

[0114] In some specific embodiments, in order to solve the above manifold optimization problem, it is assumed that the initial value of the emission data matrix X is as follows:

[0115]

[0116] where the domain of X is the manifold The manifold defined in the expression The initial Riemann gradient is The initial search direction is D0 = -G0. For the matrix X, each optimization is equivalent to optimizing the N×M elements of the matrix simultaneously.

[0117] Assume that k iterations have been completed and the current radar data matrix is ​​X k , then the Riemann gradient is Define the search function at this time as H(X k ), then the current search direction D k The mathematical expression is as follows:

[0118] D k =H(X k )=-G k +β k ·D k-1

[0119] where β k is a manually set coefficient, and its mathematical expression is as follows:

[0120]

[0121] where <·> is the Frobenius inner product on the tangent plane. β k The value of the current Riemann gradient G k and the Riemann gradient G calculated last time k-1 Related: If G k and G k-1 If the inner product of is greater than a preset threshold β0, the current coefficient β k Set to 0 to ensure the current search direction D k Same as the previous search direction D k-1 Approximately the same; however, if the difference between the two Riemann gradients is too large and the inner product is not large enough, then the update coefficient β is selected k =β n To adjust the current search direction D k , β n The expression is as follows:

[0122]

[0123] According to the current search direction D k The mathematical expression of β k The mathematical expression of β n The expression of the current search direction D k Calculation.

[0124] Further analysis of step size and error reveals that a step size that is too small will reduce the update amplitude, increasing the computational burden of the entire optimization process; a step size that is too large will prevent the algorithm from converging to the global optimal solution. Therefore, choosing an appropriate step size is crucial for an optimization algorithm.

[0125] Here we use the famous Armijo line search algorithm to calculate α k , assuming that the upper limit of the current search step size is α0, it is gradually reduced from α0 until the Armijo condition is met, so that the function can be fully reduced during the data point update process. The expression of the Armijo condition is as follows:

[0126]

[0127] in represents the directional derivative, and its expression is:

[0128]

[0129] The α of the expression that satisfies the Armijo condition k As the update step size at the kth iteration. From the expression of the Armijo condition, we can see that the control of the step size depends on the calculation of the directional derivative. Here we use Taylor expansion to discuss the first-order approximation error of the directional derivative. The function f(x) is on the tangent plane. The Taylor expansion of is as follows:

[0130]

[0131] Where t represents the step size.

[0132] Since the directional derivative is a linear term of Taylor expansion, when the step size is small, the error mainly comes from the linear term; when the step size is large, the error E(t) is mainly determined by the second-order term, and its expression is:

[0133]

[0134] Then in the logarithmic graph, theoretically, the logarithmic value of the error is linearly related to the step size t:

[0135] log(E(t))=logC+2logt

[0136] In the above formula, E(t) represents the error.

[0137] S300, constructing a remapping function to map the new data points on the tangent plane back to the manifold to achieve minimized low sidelobe waveform design.

[0138] It should be noted that, in some embodiments, step S300 may include steps S310 to S330:

[0139] S310, defining a remapping function to obtain the norm of the new data point on the tangent plane;

[0140] S320, normalizing the new data points on the tangent plane using a norm to obtain normalized new data points;

[0141] S330. Input the normalized new data points into the remapping function, multiply them by the radius of the high-dimensional hypersphere, and map the new data points on the tangent plane back to the manifold to achieve minimized low sidelobe waveform design.

[0142] In some specific embodiments, after obtaining the point x k After the Riemann gradient on the tangent plane, based on the Riemann gradient And the direction search function H(·), the current search direction d on the tangent plane can be obtained k , whose expression is:

[0143]

[0144] Finally, the original data point x k , cutting plane step α k and the search direction d on the current cutting plane k Enter a remapping function We can get a new data point x on the manifold k+1 , its expression is.

[0145]

[0146] It can be seen that the effect of the Riemannian manifold optimization algorithm depends not only on the curvature of the manifold structure, but also on the direction search algorithm and remapping algorithm selected at the corresponding data points. Different search algorithms and remapping algorithms will directly affect the optimization results based on the Riemannian manifold. The embodiment of the present invention proposes an optimization algorithm suitable for manifold structures. This method redesigns a new search function and remapping function to ensure that the proposed optimization problem can be effectively solved. Figure 5 As shown in the figure, the falling process in the Riemann manifold optimization algorithm is shown, and the remapping function is used to realize the data point from x k to x k+1 Then, the direction search function is used to update the search direction, thereby realizing iterative search.

[0147] The remap function is defined as Its expression is as follows:

[0148]

[0149] Normalize the new data matrix with its own F norm and then multiply it by the radius of the hypersphere After that, the data on the tangent plane can be projected onto the manifold Finally, we get the new data matrix X on the manifold k+1 .

[0150] Regarding the remapping function, it is necessary to focus on analyzing its non - expansiveness. The definition of non - expansiveness is as follows:

[0151]

[0152] So far, the algorithm has completed one iteration process. Whether to continue the iteration is related to the termination condition of the algorithm. In the subsequent simulation experiments, the cut - off condition of the algorithm is set as follows:

[0153] k > N p or ||G k || < g0 or α k < α0

[0154] It means that the algorithm stops iterating when it encounters one of the following situations: the current iteration number k is greater than the maximum iteration number N p , the current Riemannian gradient norm ||G k || is less than the convergence tolerance g0 of the gradient norm, and the current step size α k is less than the minimum constrained step size α0.

[0155] As Figure 6 shown, the framework of the PSL optimization algorithm based on the Riemannian manifold can be summarized into the following steps: 1) Project the Euclidean gradient onto the tangent plane of the manifold based on the projection operator to calculate the corresponding Riemannian gradient. 2) Obtain new data points in the tangent plane according to the designed direction search function. 3) Map the points in the tangent plane of the previous iteration to the manifold according to the designed remapping function, and at the same time judge whether to continue the iteration.

[0156] Verify the experimental results of the problem model based on the Riemannian manifold and the improved Riemannian conjugate gradient descent algorithm, which are mainly divided into the following three parts: Compare the relevant performance of the proposed RM method and the constant modulus performance of the output signal with other traditional waveform optimization design methods CAN and POCA respectively. Assume x = [x1, x2, …, x N T is one of the waveforms. For the CAN method, the convergence threshold ε can is set to ||x k - x k+1 || ≤ ε can = 10 -3 . For the POCA method, the convergence threshold ε poca is set to ||X new - X old ||∞ < ε poca = 10 -4 . Where X is composed of the elements in the waveform sequence x, and its expression is:

[0157] ​

[0158] Unless otherwise specified, the experimental parameters are set as follows: the number of orthogonal waveforms M = 3, the number of waveform samples of each orthogonal waveform N = 50, the parameter p of the PSL approximated by the norm expression p = 8, the PAPR upper threshold ε = 1, the maximum number of iterations N p =200, gradient norm convergence tolerance g0 = 10 -7 , the minimum constraint step size α0 = 10 -4 Weak target detection requires a waveform with better autocorrelation characteristics, because if the autocorrelation sidelobe of the waveform is too high, the weak target will be masked, thereby reducing the target detection performance. Moreover, if the cross-correlation sidelobe of the waveform is too high, the interference between the echo signals of different targets will be aggravated. Therefore, the autocorrelation weight sw a .

[0159] To further analyze the convergence of the algorithm, we first need to verify whether the defined manifold structure is suitable for solving the optimization problem PAPR(x m ).like Figure 7 The figure shows a logarithmic plot of the first-order approximation error versus step size. As can be seen from the figure, after using the manifold structure and the unconstrained optimization problem within the manifold, the rate of change of the first-order approximation error is close to the theoretical value, with only a difference in the constant term. This shows that the manifold structure used can effectively fit nonlinear spatial optimization problems.

[0160] like Figure 8 Figure 2 shows the relationship between the number of iterations and the loss function for the proposed method. It can be seen that after about 10 iterations, the loss function value drops to a level close to 0, and in subsequent iterations, the loss function value remains almost unchanged, indicating that the current parameter selection can effectively achieve convergence of the optimization algorithm.

[0161] like Figure 9 The figure shows the effect of the actual number of iterations of the proposed RM optimization algorithm on the autocorrelation function of the waveform transmission sequence. Only the case where the number of antennas M = 1 is considered here, and the other parameters of the optimization algorithm are set to ensure that it can complete the specified number of iterations. As can be seen from the figure, the normalized sidelobe level of the ACF decreases overall with increasing iteration numbers. When the number of iterations is equal to 10, 20, and 50, the PSL and ISL levels of the autocorrelation function of the transmitted waveform sequence are both high. As the number of iterations increases, the sidelobe level of the autocorrelation function of the transmitted waveform sequence also decreases. However, when the number of iterations exceeds 100, increasing the number of iterations has no significant effect on optimizing the sidelobe level of the transmitted waveform. Therefore, for the proposed RM optimization algorithm, setting the maximum number of iterations to 200 is appropriate.

[0162] like Figure 10 as well as Figure 11As shown in Figure 2, the convergence tolerance of the gradient norm and the minimum constraint step size will affect the iterative process of the optimization algorithm, and thus affect the optimization effect of PSL. For the case of the transmission sequence length N = 50, when the convergence tolerance of the gradient norm drops to g0 = 10 -4 Before, the PSL of the correlation function of the output signal dropped rapidly. This result shows that choosing a suitable convergence margin can significantly reduce the sidelobe level of the optimized waveform. -5 At the beginning, the PSL gradually stabilizes. When g0 continues to decrease, it even fluctuates slightly, indicating that the marginal benefit of overly strict convergence tolerance on performance improvement gradually decreases, which may be limited by numerical accuracy or local minimum. This shows that the convergence tolerance of the gradient norm has a limited impact on the PSL of the output signal. It is set to g0 = 10 -4 is a reasonable value. The effect of the minimum constraint step size on PSL also shows a similar trend. When α0 changes from 10 -1 Reduced to 10 -3 When α0 is less than 10 -5 After , the improvement of PSL slows down, indicating that the algorithm converges early.

[0163] Further correlation and robustness analysis is carried out, such as Figure 12 The following is a comparison of correlation functions for sequence length N = 50 and different peak-to-average ratio constraints (PAPR = 1 and PAPR = 8). The experiment considers the effects of strict constant modulus constraints and looser constant modulus constraints on the side lobes of the correlation function.

[0164] exist Figure 12 In (a), the autocorrelation PSL and ISL levels of the design based on the CAN method are both the highest, with the PSL close to -20dB; the autocorrelation PSL based on the RM method is around -30dB, and the overall sidelobe fluctuation is small; the autocorrelation PSL based on the POCA method is around -37dB, and its ISL level is also low. However, in the subsequent constant modulus performance analysis, the constant modulus performance of its output waveform is poor, which easily leads to the problem of output signal distortion when the signal enters the nonlinear region of the power amplifier. Figure 12 In (b), the cross-correlation PSL designed based on the POCA method is the highest, at around -12dB. However, as the cross-correlation delay increases, the cross-correlation sidelobe level gradually decreases, and the overall sidelobe level is lower than that of the CAN method. The cross-correlation PSL designed based on the CAN method is around -19dB, and as the cross-correlation delay increases, its sidelobe drop is not as obvious as that of the POCA method. The cross-correlation PSL designed using the RM method is around -44dB, which is significantly better than the first two methods, and the overall sidelobe level is also lower. Figure 12In (c), due to the increase in PAPR, the waveform no longer meets the strict constant modulus constraint. At this time, the autocorrelation PSL of the RM-based design method is about -60dB, and the overall autocorrelation sidelobe level is lower than that of the other two methods. Figure 12 In (d), due to the improvement of PAPR, the cross-correlation sidelobe level of the waveform obtained by the RM-based method decreases significantly as the cross-correlation delay increases.

[0165] like Figure 13 As shown, it is a comparison diagram of correlation functions under different peak-to-average ratio constraints (PAPR=1 and PAPR=8) with sequence length N=100. Figure 13 (a) shows the ACF comparison of PAPR=1. Figure 13 (b) shows the CCF comparison of PAPR=1. Figure 13 (c) shows the ACF comparison of PAPR=8. Figure 13 (d) in the figure shows the CCF comparison when PAPR=8. Figure 13 In (a), when the sequence length N = 100, compared with Figure 12 In the case of N = 50 in (a), the autocorrelation PSL and ISL of the output waveforms obtained by all optimization methods decrease, indicating that improving the waveform sequence increases the degree of freedom of the optimization algorithm's parameters. When PAPR = 1, the autocorrelation PSL of the waveform optimized by the CAN method is approximately -22dB, the autocorrelation PSL of the output waveform optimized by the RM method is approximately -33dB, and the autocorrelation PSL of the output waveform optimized by the POCA method is approximately -50dB. When the PAPR is increased, the overall sidelobe level of the autocorrelation function of the RM optimization algorithm decreases rapidly, with a PSL of approximately -63dB. As will be seen in the subsequent constant modulus performance analysis section, the improvement in correlation performance comes at the expense of constant modulus performance.

[0166] like Figure 14 As shown, for the cross-correlation function when PAPR=1, Figure 14 This is a comparison of the correlation functions when the sequence length is N = 200, PAPR = 1 and PAPR = 8, where Figure 14 (a) shows the ACF comparison of PAPR=1. Figure 14 (b) shows the CCF comparison of PAPR=1. Figure 14 (c) shows the ACF comparison of PAPR=8. Figure 14 (d) in FIG. 5 shows the CCF comparison when PAPR=8.

[0167] Finally, the constant modulus performance analysis is carried out, where Figure 15 The output waveform modulus under different PAPRs is shown as N=50 and N=100. Figure 15 The modulus of the output waveform sequence with different sequence lengths (50 and 100) under different PAPRs was investigated.

[0168] Figure 15 (a) and Figure 15 As shown in (b), when the sequence lengths are 50 and 100, respectively, the real and imaginary parts of the output signals obtained using the CAN and RM optimization methods form a unit circle, indicating that the output waveforms have good constant modulus properties. This is because the CAN algorithm directly optimizes the phase-coded signal, while the RM algorithm proposed in this paper dynamically projects the PAPR upper threshold ε onto the unit complex circle manifold. However, for the output waveforms obtained using the POCA method, the real and imaginary parts are scattered across sequences of different lengths, indicating that their amplitudes vary significantly and fail to meet the constant modulus constraint. These results demonstrate that the POCA method achieves lower sidelobe levels by modifying the constant modulus constraint.

[0169] Figure 15 (c) and Figure 15 As shown in (d) of the figure, when the upper threshold ε of the RM algorithm is no longer 1, the real and imaginary parts of the output waveform no longer form a unit circle. Compared with the POCA method, which also does not conform to the unit circle, the energy distribution is more concentrated, and the modulus distribution mostly lies within the unit circle. The output waveform of the CAN algorithm still maintains a strict constant modulus constraint.

[0170] In summary, the embodiments of the present invention have the following improvements compared to the prior art:

[0171] 1) Using the Riemannian manifold optimization method, the original NP-hard constrained problem is transformed into an unconstrained problem in the manifold space, effectively avoiding the computational complexity brought by constraint processing.

[0172] 2) The Riemannian manifold is used to search for waveforms that meet the requirements from specific structures, which improves the convergence speed and solution accuracy of the algorithm.

[0173] 3) The present invention can design a low-correlation sidelobe waveform matrix with good orthogonal characteristics under specified PAPR constraints.

[0174] Therefore, the embodiment of the present invention introduces the manifold structure in Riemannian geometry into the non-convex waveform design optimization problem, and proposes a low-correlation sidelobe waveform matrix design method based on Riemannian manifold for minimizing the waveform sidelobe problem model and PAPR constraint conditions. Compared with the existing technology, although the low-sidelobe waveform design using neural networks can effectively accelerate the convex optimization algorithm, its optimization effect is quite dependent on the setting of the regularization term of the loss function. When faced with complex scenarios such as MIMO waveform design with constant modulus constraints and orthogonal constraints, it is difficult to ensure that the output waveform results strictly meet the constraints. Traditional waveform optimization methods and waveform optimization methods based on neural networks both rely on specific signal models, but these specific signal models may deviate from the optimal waveform space. The Riemannian manifold method can reduce computational complexity and effectively improve the algorithm convergence speed and solution accuracy.

[0175] See also Figure 2 The present application also provides a Riemannian manifold-based low sidelobe waveform design system, which can implement the above-mentioned Riemannian manifold-based low sidelobe waveform design method. The system includes:

[0176] The first module 201 is used to introduce the peak-to-average power ratio as a constraint condition based on the MIMO radar system to construct an unconstrained optimization problem of a manifold, where the manifold is a high-dimensional hypersphere in a complex space with a preset radius length;

[0177] The second module 202 is used to introduce a tangent plane and perform an iterative search on the unconstrained optimization problem of the manifold to obtain new data points on the tangent plane;

[0178] The third module 203 is used to construct a remapping function to map the new data points on the tangent plane back to the manifold to achieve a minimized low sidelobe waveform design.

[0179] It can be understood that the contents of the above method embodiments are all applicable to the present system embodiments, the functions specifically implemented by the present system embodiments are the same as those of the above method embodiments, and the beneficial effects achieved are also the same as those achieved by the above method embodiments.

[0180] The preferred embodiments of the present invention are described above with reference to the accompanying drawings, but are not intended to limit the scope of the present invention. Any modifications, equivalent substitutions, and improvements made by those skilled in the art without departing from the scope and essence of the present invention should be within the scope of the present invention.

Claims

1. A low sidelobe waveform design method based on Riemannian manifold, characterized in that: The method comprises the following steps: Based on a MIMO radar system, the peak-to-average power ratio is introduced as a constraint to construct an unconstrained optimization problem on a manifold, which is a high-dimensional hypersphere in complex space with a preset radius. A tangent plane is introduced, and an unconstrained optimization problem of the manifold is iteratively searched to obtain new data points on the tangent plane; A remapping function is constructed to map the new data points on the tangent plane back to the manifold to achieve a minimized low sidelobe waveform design.

2. The method according to claim 1, characterized in that Based on the MIMO radar system, the peak-to-average power ratio is introduced as a constraint condition to construct an unconstrained optimization problem of the manifold, including: Based on a MIMO radar system, a transmission data matrix of the MIMO radar system is obtained and constant modulus constraints are performed to obtain an ideal transmission data matrix; Defining a minimization criterion of a correlation function of the ideal transmission data matrix as an optimization criterion for waveform design, defining a norm of the ideal transmission data matrix as a measure for minimizing a peak sidelobe level, and constructing a peak sidelobe level function for minimizing a transmission waveform; The peak sidelobe level function of the minimized transmission waveform is transformed by using the peak-to-average power ratio as a constraint condition to obtain a constrained nonlinear optimization problem; The feasible solution of the peak sidelobe level function of the minimized transmission waveform is defined in a manifold obtained by multiplying a plurality of smooth Riemannian manifolds, and the constrained nonlinear optimization problem is converted into an unconstrained optimization problem of the manifold.

3. The method according to claim 2, characterized in that The expression of the peak sidelobe level function of the minimized transmission waveform is specifically as follows: In the above formula, f(X) represents the norm of the ideal transmission data matrix, X represents the ideal transmission data matrix, and w a 、w c represents the weight coefficient, represents the autocorrelation function level of all antennas, represents the cross-correlation function level between all antennas, N k represents a matrix that changes with the change of the delay variable k, k represents the delay variable, x i 、x j Represents the transmitted waveform sequence, M represents the number of transmitting antennas, and N represents the number of waveform samples collected by each antenna. Represents x i The conjugate transpose of , p represents the power of the norm.

4. The method according to claim 2, characterized in that The expression of the unconstrained optimization problem of the manifold is specifically as follows: In the above formula, represents the manifold, X represents the ideal emission data matrix, f(X) represents the norm of the ideal emission data matrix, represents an element in the manifold, x m represents the sequence transmitted by the mth antenna, represents the complex space, N represents the number of waveform samples collected by each antenna, ε represents the threshold corresponding to different levels of peak-to-average power ratio, m represents the mth antenna, and M represents the number of transmitting antennas.

5. The method according to claim 1, wherein The introducing of the tangent plane and performing iterative search on the unconstrained optimization problem of the manifold to obtain new data points on the tangent plane include: A tangent plane is defined, and the unconstrained optimization problem of the manifold is linearized through the tangent plane to obtain data points of the tangent plane, where the tangent plane is a vector space composed of tangent vectors of all manifolds passing through the points; On the tangent plane, calculating the current Riemann gradient for the data point to determine the fastest descent direction; Determine a direction search function based on the current Riemann gradient and determine a step size using an Armijo line search algorithm; The direction search function is used to convert the current Riemann gradient into a specific search direction to search the unconstrained optimization problem of the manifold, and iterative update is performed in combination with the step size to obtain new data points on the tangent plane.

6. The method according to claim 5, characterized in that The expression for determining the step size by the Armijo line search algorithm is specifically as follows: In the above formula, X k represents the new data point on the tangent plane, α k Indicates the current step size, D k Indicates the current search direction. represents the directional derivative, G k Conjugate transpose, G k represents the Riemann gradient of the kth iteration, σ represents the update coefficient, X k represents the waveform matrix of the kth iteration, f(X k ) represents the expression of the peak sidelobe level function, f(X k +α k D k ) represents the updated peak sidelobe level.

7. The method according to claim 5, characterized in that The expression of the direction search function is specifically as follows: D k =H(X k )=-G k +β k ·D k-1 In the above formula, D k Indicates the current search direction, H(X k ) represents the search function, G k represents the Riemann gradient, β k Denotes the coefficient, D k-1 Indicates the last search direction.

8. The method according to claim 1, characterized in that The constructing of the remapping function maps the new data points on the tangent plane back to the manifold to achieve a minimized low sidelobe waveform design, including: defining a remapping function to obtain the norm of the new data point on the tangent plane; Normalizing the new data points on the tangent plane using the norm to obtain normalized new data points; The normalized new data points are input into the remapping function, multiplied by the radius of the high-dimensional hypersphere, and the new data points on the tangent plane are mapped back to the manifold to achieve minimized low sidelobe waveform design.

9. The method according to claim 8, characterized in that The expression of the remapping function is specifically as follows: In the above formula, represents the remapping function, X k represents the new data point on the tangent plane, α k Indicates the current step size, D k Indicates the current search direction. represents the radius of the high-dimensional hypersphere, ||·|| F Represents the norm.

10. A low sidelobe waveform design system based on Riemannian manifold, characterized in that: The system comprises: The first module is used to construct an unconstrained optimization problem of a manifold based on a MIMO radar system by introducing the peak-to-average power ratio as a constraint condition. The manifold is a high-dimensional hypersphere in a complex space with a preset radius length. The second module is used to introduce a tangent plane, perform an iterative search on the unconstrained optimization problem of the manifold, and obtain new data points on the tangent plane; The third module is used to construct a remapping function to map the new data points on the tangent plane back to the manifold to achieve minimized low sidelobe waveform design.

Images